Prove f: Zmod p---> Z mod p is a ring homomorphism and find the kernel of f.

Find all possible non-trivial homomorphism

a) S₃ --> Z₆

b) D₄ --> Z₄

c) Z₅ --> Z₁₀

d) Z₁₀ --> Z₅

Find polynomial of degree at most 1 best approximates function f(x) = e^x in the sense of uniform approximation on [0,1]. give the error of approximation of the function f with this polynomial.

Consider the IVP x' = t^2 +x^2, x(0) = 1. Complete the following table for the numerical solutions of given IVP with step-size h = 0.05.

t - x by Euler’s Method - x by Improved Euler’s Method

0 -    1 - 1

0.05 - …….    - ……...

0.1 -    ……. - ……..

(Symmetric Matrices and Quadratic Forms) Examine the following functions for relative extremum values: (a) F = x2y + 2x2 − 2xy + 3y2 − 4x + 7y (b) F = x3 + 4x2 + 3y2 + 5x − 6y

Wе sаw in сlаss thаt in bipаrtitе grаphs thе mаximum mаtсhing аnd minimum vеrtеx сovеr hаvе thе sаmе sizе. (Thе numbеr of еdgеs in thе mаtсhing еquаls thе numbеr of vеrtiсеs in thе сovеr.)

(а)Find аn еxаmplе of а non-bipаrtitе grаph in whiсh thе minimum vеrtеx сovеr is еxасtly twiсе аs lаrgе аs thе mаximum mаtсhing.

(b)Find аnothеr еxаmplе of а non-bipаrtitе grаph in whiсh thе minimum vеrtеx сovеr аnd thе mаximum mаtсhing hаvе еquаl sizеs.

Find y as a function of x if

y(4)−12y‴+36y″=−512e^-2x

y(0)=−1,  y′(0)=10,  y″(0)=28,  y‴(0)=16

y(x)=

A mass of 2 kg is suspended from a spring with known spring constant of 10N/m and allowed to come to rest. it is then set in motion by giving it an initial velocity of 150 cm/sec. Find an expression for the motion of the mass, assuming no air resistance.

If X is any topological space, a subset A ⊆ X is compact (in the subspace topology) if and only if every cover of A by open subsets of X has a finite subcover.

Find two linearly independent power series solutions of the given differential equation about the ordinary point x=0.

y''-2xy=0

Consider the equation u_tt = u_xx

x ∈ (0, pi)

u_x(0,t) = u(pi,t) = 0

Write the series expansion for a solution u(x,t)

The price of a home in Medford was $ 100,000 in 1985 and rose to $ ⁢ 170 ,000 in 1999 .
a. Create two models, f ( t ) assuming linear growth and     g ( t ) assuming exponential growth, where t is the number of years after 1985 . Round coefficients to three decimal places when necessary. f ( t ) =     g ( t ) =

b. Fill in the following table using the equations you just found. Use the exact equation for exponential growth.
Round your answers to the nearest integer.

1) A radioactive substance decays at a rate proportional to the amount of the substance at present time. Initially 200 grams of a the substance was present and remain 80% of the initial amount after 2 hours.

A.) Determine the amount of the substance remaining after 10 hours (counted in grams)

B.) Determine the time that 60% of the initial amount of the substance has decayed (counted in hours)